Question

The number of distinct real roots of equation $3x^4+4x^3-12x^2+4=0$ is

Answer 1

Let $f\left(x\right)=3x^4+4x^3-12x^2+4$
differentiation w.r.t. $x,$ we get
$\Rightarrow f^{\prime }\left(x\right)=12x^3+12x^2-24x$
$\Rightarrow f^{\prime }\left(x\right)=12x(x+2)(x-1)$
For max/min, $f^{\prime }\left(x\right)=0$
$\Rightarrow x=0,-2,1$
$\Rightarrow f\left(0\right)=4,f(-2)=-28,f\left(1\right)=-1$
So, using intermediate value theorem on the given function, $4$ real roots exist.